The local functors of points of Supermanifolds

We study the local functor of points (which we call the Weil-Berezin functor) for smooth supermanifolds, providing a characterization, representability theorems and applications to differential calculus

Highest weight Harish-Chandra supermodules and their geometric realizations

In this paper we discuss the highest weight $\frak k_r$-finite representations of the pair $(\frak g_r,\frak k_r)$ consisting of $\frak g_r$, a real form of a complex basic Lie superalgebra of classical type $\frak g$ (${\frak g}\neq A(n,n)$), and the maximal compact subalgebra $\frak k_r$ of $\frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.Comment: This article contains of part of the material originally posted as arXiv:1503.03828 and arXiv:1511.01420. The rest of the material was posted as arXiv:1801.07181 and will also appear in an enlarged version as subsequent postin

SUSY structures, representations and Peter-Weyl theorem for S1∣1S^{1|1}

The real compact supergroup $S^{1|1}$ is analized from different perspectives and its representation theory is studied. We prove it is the only (up to isomorphism) supergroup, which is a real form of $({\mathbf C}^{1|1})^\times$ with reduced Lie group $S^1$, and a link with SUSY structures on ${\mathbf C}^{1|1}$ is established. We describe a large family of complex semisimple representations of $S^{1|1}$ and we show that any $S^{1|1}$-representation whose weights are all nonzero is a direct sum of members of our family. We also compute the matrix elements of the members of this family and we give a proof of the Peter-Weyl theorem for $S^{1|1}$

Super Distributions, Analytic and Algebraic Super Harish-Chandra pairs

The purpose of this paper is to extend the theory of Super Harish-Chandra pairs, originally developed by Koszul for Lie supergroups, to analytic and algebraic supergroups, in order to obtain information also about their representations. We also define the distribution superalgebra for algebraic and analytic supergroups and study its relation with the universal enveloping superalgebr

Testing Cosmological General Relativity against high redshift observations

Several key relations are derived for Cosmological General Relativity which are used in standard observational cosmology. These include the luminosity distance, angular size, surface brightness and matter density. These relations are used to fit type Ia supernova (SNe Ia) data, giving consistent, well behaved fits over a broad range of redshift $0.1 < z < 2$. The best fit to the data for the local density parameter is $\Omega_{m} = 0.0401 \pm 0.0199$. Because $\Omega_{m}$ is within the baryonic budget there is no need for any dark matter to account for the SNe Ia redshift luminosity data. From this local density it is determined that the redshift where the universe expansion transitions from deceleration to acceleration is $z_{t}= 1.095 {}^{+0.264}_{-0.155}$. Because the fitted data covers the range of the predicted transition redshift $z_{t}$, there is no need for any dark energy to account for the expansion rate transition. We conclude that the expansion is now accelerating and that the transition from a closed to an open universe occurred about $8.54 {\rm Gyr}$ ago.Comment: Rewritten, improved and revised the discussion. This is now a combined paper of the former version and the Addendu

HIGHEST WEIGHT HARISH-CHANDRA SUPERMODULES AND THEIR GEOMETRIC REALIZATIONS

In this paper we discuss the highest weight kr-finite representations of the pair (r, kr) consisting of r, a real form of a complex basic Lie superalgebra of classical type ( 60 A(n, n)), and the maximal compact subalgebra kr of r,0, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces Gr/Kr